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Study Guides > MTH 163, Precalculus

Identify polynomial functions

An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius r of the spill depends on the number of weeks w that have passed. This relationship is linear.

[latex]r\left(w\right)=24+8w\\[/latex]

We can combine this with the formula for the area A of a circle.

[latex]A\left(r\right)=\pi {r}^{2}\\[/latex]

Composing these functions gives a formula for the area in terms of weeks.

[latex]\begin{cases}A\left(w\right)=A\left(r\left(w\right)\right)\\ =A\left(24+8w\right)\\ =\pi {\left(24+8w\right)}^{2}\end{cases}\\[/latex]

Multiplying gives the formula.

[latex]A\left(w\right)=576\pi +384\pi w+64\pi {w}^{2}\\[/latex]

This formula is an example of a polynomial function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

A General Note: Polynomial Functions

Let n be a non-negative integer. A polynomial function is a function that can be written in the form

[latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}\\[/latex]

This is called the general form of a polynomial function. Each [latex]{a}_{i}\\[/latex] is a coefficient and can be any real number. Each product [latex]{a}_{i}{x}^{i}\\[/latex] is a term of a polynomial function.

Example 4: Identifying Polynomial Functions

Which of the following are polynomial functions?

[latex]\begin{cases}f\left(x\right)=2{x}^{3}\cdot 3x+4\hfill \\ g\left(x\right)=-x\left({x}^{2}-4\right)\hfill \\ h\left(x\right)=5\sqrt{x}+2\hfill \end{cases}\\[/latex]

Solution

The first two functions are examples of polynomial functions because they can be written in the form [latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}\\[/latex], where the powers are non-negative integers and the coefficients are real numbers.

  • [latex]f\left(x\right)\\[/latex] can be written as [latex]f\left(x\right)=6{x}^{4}+4\\[/latex].
  • [latex]g\left(x\right)\\[/latex] can be written as [latex]g\left(x\right)=-{x}^{3}+4x\\[/latex].
  • [latex]h\left(x\right)\\[/latex] cannot be written in this form and is therefore not a polynomial function.

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  • Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..